Abstract
We consider the existence and uniqueness of singular solutions for equations of the formu 1=div(|Du|p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
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Dedicated to Professor Shmuel Agmon
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Kamin, S., Vazquez, J.L. Singular solutions of some nonlinear parabolic equations. J. Anal. Math. 59, 51–74 (1992). https://doi.org/10.1007/BF02790217
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DOI: https://doi.org/10.1007/BF02790217